Once a homogeneous sample of superclusters or individual voids is
identified, their structural properties can be determined and then
studied statistically, just as is done for other cosmic objects (e.g.
stars, supernova remnants, galaxies); in 1981, Oort
(127) pointed the
way. The locational properties of the galaxies or clusters that
constitute the homogeneous data bases used to identify voids can also
be studied by statistical methods to learn about void properties and
their relation to predictions of models; the techniques and results
are described below.

3.1.1. POISSON VOIDS (SUBTLETIES) Individual voids
in the space distribution of galaxies with
characteristic lengths of ~ 50 Mpc were first recognized definitively
by visual inspection (Sections 1.3,
2.1.1); the Coma and Hercules
voids are illustrated in Figure
4. Bahcall & Soneira
(15) presented a
variety of evidence that suggests the physical presence of a void in
the space distribution of the 71 northern Abell clusters of galaxies
(statistical sample, distance classes D 4) corresponding to
Nt 52
superclusters distributed according to the selection function
f (b) = 100.3(1 - csc b) (where b is
Galactic latitude) in solid angle
t The void
was detected visually as an empty region of solid angle
v =
gvt
(where gv 1/7) in the surface
distribution of Abell clusters. From
an analysis of 100 computer model simulations and an analytical
calculation for a model of Nt superclusters with uniformly
Poisson-distributed locational coordinates, they estimated that the
statistical probability for the chance occurrence of a void of solid
angle
>
v is
Pv
0.01. Politzer &
Preskill [147; see also
(135)]
proved that a void search-procedure correction must be applied to
Bahcall & Soneira's analytical calculation [to allow for the fact that
Bahcall & Soneira identified the void by scanning the survey area
t
to find the largest empty region,
v, and
not by placing search windows of solid angle
v at
random (or, alternatively, evenly spaced) locations within
t]. Hence, Bahcall & Soneira's probability
formula
Pvt /
v ×
e-Ntgv became replaced
(for circular voids) by Politzer & Preskill's
Pvt
/ v
× (Ntgv)2 ×
e-Ntgv / fc
(where fc is a fiducial correction factor to
exclude circles that overlap the boundary of the sample), so that
Pv ~
0.2. The value is even larger if the shape of the void is allowed
additional degrees of freedom [see
(147)]. The factor of
> 20
discrepancy between the value of Pv derived from
computer model simulations and the value derived analytically is
unexplained at this time.